Entrainment and detrainment in a simple cumulus cloud model

A cumulus cloud model, analogous to the mixed-layer models of the planetary boundary layer and the upper ocean, is developed using a single, unitary entrainment process in which the motion of the cloud boundary relative to the mean flow is permitted, produced, and controlled by turbulent processes. An alternate theory to the mixing-length theory of Asai and Kasahara (1967) is proposed which completely removes the strong scale-dependence of the Asai-Kasahara model. The model reintroduces scale-dependence by introducing including the pe5turbation pressure term of the equation of vertical motion. It is shown that the model predicts deeper clouds than the Asai-Kasahara model for a given sounding, due to the entrainment assumption and the effects of the perturbation pressure. Lateral entrainment dominates cloud-top entrainment, although finite-difference errors increase the cloud-top entrainment rate from zero to a positive value in actual situations. The fractional entrainment rate for updrafts is determined to vary only slightly with height and to decrease only slowly as the cloud radius increases, while the fractional detrainment rate for updrafts increases with height.